Calculus Examples

Find dy/dx y = natural log of ((4x^2)/(5x^5+4))^4
Step 1
Differentiate both sides of the equation.
Step 2
The derivative of with respect to is .
Step 3
Differentiate the right side of the equation.
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Step 3.1
Differentiate using the chain rule, which states that is where and .
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Step 3.1.1
To apply the Chain Rule, set as .
Step 3.1.2
The derivative of with respect to is .
Step 3.1.3
Replace all occurrences of with .
Step 3.2
Differentiate using the chain rule, which states that is where and .
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Step 3.2.1
To apply the Chain Rule, set as .
Step 3.2.2
Differentiate using the Power Rule which states that is where .
Step 3.2.3
Replace all occurrences of with .
Step 3.3
Differentiate using the Constant Multiple Rule.
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Step 3.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.3.2
Multiply by .
Step 3.4
Differentiate using the Quotient Rule which states that is where and .
Step 3.5
Differentiate.
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Step 3.5.1
Differentiate using the Power Rule which states that is where .
Step 3.5.2
Move to the left of .
Step 3.5.3
By the Sum Rule, the derivative of with respect to is .
Step 3.5.4
Since is constant with respect to , the derivative of with respect to is .
Step 3.5.5
Differentiate using the Power Rule which states that is where .
Step 3.5.6
Multiply by .
Step 3.5.7
Since is constant with respect to , the derivative of with respect to is .
Step 3.5.8
Simplify the expression.
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Step 3.5.8.1
Add and .
Step 3.5.8.2
Multiply by .
Step 3.6
Multiply by by adding the exponents.
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Step 3.6.1
Move .
Step 3.6.2
Use the power rule to combine exponents.
Step 3.6.3
Add and .
Step 3.7
Combine and .
Step 3.8
Move to the left of .
Step 3.9
Simplify.
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Step 3.9.1
Apply the product rule to .
Step 3.9.2
Apply the product rule to .
Step 3.9.3
Apply the product rule to .
Step 3.9.4
Apply the product rule to .
Step 3.9.5
Apply the distributive property.
Step 3.9.6
Apply the distributive property.
Step 3.9.7
Apply the distributive property.
Step 3.9.8
Combine terms.
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Step 3.9.8.1
Raise to the power of .
Step 3.9.8.2
Multiply the exponents in .
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Step 3.9.8.2.1
Apply the power rule and multiply exponents, .
Step 3.9.8.2.2
Multiply by .
Step 3.9.8.3
Multiply by the reciprocal of the fraction to divide by .
Step 3.9.8.4
Multiply by .
Step 3.9.8.5
Multiply by .
Step 3.9.8.6
Raise to the power of .
Step 3.9.8.7
Use the power rule to combine exponents.
Step 3.9.8.8
Add and .
Step 3.9.8.9
Multiply by .
Step 3.9.8.10
Multiply by .
Step 3.9.8.11
Multiply by .
Step 3.9.8.12
Multiply by .
Step 3.9.8.13
Subtract from .
Step 3.9.8.14
Raise to the power of .
Step 3.9.8.15
Multiply the exponents in .
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Step 3.9.8.15.1
Apply the power rule and multiply exponents, .
Step 3.9.8.15.2
Multiply by .
Step 3.9.8.16
Multiply by .
Step 3.9.8.17
Multiply by by adding the exponents.
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Step 3.9.8.17.1
Use the power rule to combine exponents.
Step 3.9.8.17.2
Add and .
Step 3.9.8.18
Move to the left of .
Step 3.9.8.19
Multiply by .
Step 3.9.8.20
Move to the left of .
Step 3.9.8.21
Cancel the common factor of and .
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Step 3.9.8.21.1
Factor out of .
Step 3.9.8.21.2
Cancel the common factors.
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Step 3.9.8.21.2.1
Factor out of .
Step 3.9.8.21.2.2
Cancel the common factor.
Step 3.9.8.21.2.3
Rewrite the expression.
Step 3.9.8.22
Cancel the common factor of and .
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Step 3.9.8.22.1
Factor out of .
Step 3.9.8.22.2
Cancel the common factors.
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Step 3.9.8.22.2.1
Factor out of .
Step 3.9.8.22.2.2
Cancel the common factor.
Step 3.9.8.22.2.3
Rewrite the expression.
Step 3.9.8.23
Cancel the common factor of and .
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Step 3.9.8.23.1
Factor out of .
Step 3.9.8.23.2
Cancel the common factors.
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Step 3.9.8.23.2.1
Factor out of .
Step 3.9.8.23.2.2
Cancel the common factor.
Step 3.9.8.23.2.3
Rewrite the expression.
Step 3.9.8.24
Cancel the common factor of and .
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Step 3.9.8.24.1
Factor out of .
Step 3.9.8.24.2
Cancel the common factors.
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Step 3.9.8.24.2.1
Factor out of .
Step 3.9.8.24.2.2
Cancel the common factor.
Step 3.9.8.24.2.3
Rewrite the expression.
Step 3.9.8.25
Cancel the common factor of and .
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Step 3.9.8.25.1
Factor out of .
Step 3.9.8.25.2
Cancel the common factors.
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Step 3.9.8.25.2.1
Cancel the common factor.
Step 3.9.8.25.2.2
Rewrite the expression.
Step 3.9.9
Reorder terms.
Step 3.9.10
Simplify the numerator.
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Step 3.9.10.1
Factor out of .
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Step 3.9.10.1.1
Factor out of .
Step 3.9.10.1.2
Factor out of .
Step 3.9.10.1.3
Factor out of .
Step 3.9.10.2
Rewrite the expression using the negative exponent rule .
Step 3.9.10.3
Combine and .
Step 3.9.10.4
To write as a fraction with a common denominator, multiply by .
Step 3.9.10.5
Combine and .
Step 3.9.10.6
Combine the numerators over the common denominator.
Step 3.9.10.7
Combine exponents.
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Step 3.9.10.7.1
Combine and .
Step 3.9.10.7.2
Combine and .
Step 3.9.10.8
Remove unnecessary parentheses.
Step 3.9.10.9
Reduce the expression by cancelling the common factors.
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Step 3.9.10.9.1
Factor out of .
Step 3.9.10.9.2
Factor out of .
Step 3.9.10.9.3
Cancel the common factor.
Step 3.9.10.9.4
Rewrite the expression.
Step 3.9.11
Multiply the numerator by the reciprocal of the denominator.
Step 3.9.12
Multiply by .
Step 3.9.13
Factor out of .
Step 3.9.14
Rewrite as .
Step 3.9.15
Factor out of .
Step 3.9.16
Rewrite as .
Step 3.9.17
Move the negative in front of the fraction.
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Replace with .